Linear Algebra Help: Homework Equations & Solutions

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Homework Statement

There are two questions in my problem set that are giving me a hard time:
1. The cross product of two vectors in R^3 is defined by [a1, a2, a3]X[b1, b2, b3]=[a2b3-a3b2, a3b1-a1b3, a1b2-a2b1] (they are all one column) Consider an arbitrary vector v in R^3. Is the transformation T(x)=VxX from r^3 to R^3 linear? If so, find its matrix in terms of the components of the vector v.
2. Find matrices of the linear transformations from R^3 to R^3:
The rotation about the y-axis through an angle theta, clockwise as viewed from the positive y-axis.

Homework Equations

The Attempt at a Solution

1. In this one I pretty much crossed v and x to get [v2x3-v3x2, v3x1-v1x3, v1x2-v2x1] after this I'm not too sure. I also only know that it is linear if the x matrix is scaled. (I think I'm wrong about that) I pretty sure it would help if I know what "linear" mean in this context.
2. A little question about the wording, is it saying that it is reflected about the y axis? If so would it be [-1 0 0, 0 1 0, 0 0 -1]?

Thanks in advance.

 

[HallsofIvy]

"Linear" means that T(u+ w)= T(u)+ T(w) and that T(au= aT(u) where u and w are vectors and a is a number.

Is $(\vec{u}+ \vec{w})\times\vec{v}= \vec{u}\times\vec{v}+ \vec{w}\vec{v}$? Is $(a\vec{u})\times\vec{v}= a(\vec{u}\times\vec{v})$?